FENG Ting-fu,ZHANG Ke-lei

(1.School of Mathematics,Kunming University,Kunming,Yunnan,650214)

(2.School of Mathematics and Computating Sciences,Guilin University of Electronic Science and Technology,Guilin,Guangxi,541004)

Abstract:The integral functional of Hrmander’s vector fields is considered,by virtue of the Sobolev inequality related to Hrmander’s vector fields and the iteration formula of Stampacchia,it is proved that the minimizers of integral functional have higher integrability with the boundary data allowing the higher integrability.Moreover,the L1(Ω)and L∞(Ω)boundedness of minimizers are also given,which extends the results of Leonetti and Siepe[12]and Leonetti and Petricca[13]from Euclidean spaces to Hrmander’s vector fields.

Keywords: Hrmander’s vector fields;Integral functional;Minimizers;Integrability;Boundedness

1 Introduction

We consider the integral functional of Hormander’s vector fields

where Ω⊂n(n≥3)is a bounded open set,X={X1,···,Xm}(m≥n)areC∞vector fields in Ω satisfying the Hrmander’s finite rank condition[11],rank Lie[X1,···,Xm]=n,where,···,m.Note that,whenf(x,z)in(1.1)is a Carath´eodory function and satis fies the standard growth condition|z|p≤f(x,z)≤c(1+|z|p),1

by direct method and obtained Hlder continuity by Moser’s method.Furthermore,Xu[17]obtainedC∞continuity by similar method.Afterwards,Giannetti[7]obtained higher integrability of the minimizers of(1.1)under the growth condition

In this paper we assume thatf(x,z)in(1.1)is a Carathodory function and satis fies the standard growth condition

2 Main Results and Preliminary Knowledge

De finition 2.1[3,6]For any 1

for anyt0>0 and some positive constantsc=c(f),wheremeasEdenotes thendimensional Lebesgue measure ofE⊂n.Iff∈(Ω),thenf∈Lq0(Ω)for any 1≤q0

In this paper,our mian results are sated as follows.

Inspired by Leonetti and Siepe[12],for a minimizeruof(1.1)with the condition(1.2),we can rewriteuasu=u∗+(u−u∗),our aim is to prove when the boundary datumu∗has the higher integrability,u−u∗also has the higher integrability.The following two lemmas are needed for the proof of Theorem 2.4.

Lemma 2.5[3,6]Let Ω⊂nbe a bounded open set.Then for anyu∈(Ω),1

ifβ=1,then

3 Proof of Theorem 2.4

Proof of Theorem 2.4For anyk∈(0,+∞),suppose thatTk:→is a function such that

settingψ=u−u∗−Tk(u−u∗),it follows from(3.1)that

where 1A(x)=1 ifx∈A,1A(x)=0 ifx/∈A.Let us consider

Combining(2.2),(3.6)and Ω ={|u−u∗|≤k}∪{|u−u∗|>k},it concludes

and then by(3.7),

It follows from(1.2),(3.3),(3.8)and Lemma 2.5 that

Sincep

Finally we insert(3.11)into(3.10),we easily obtain

For anyh>k≥k0,it follows from(3.4)that

Combining(3.12)and(3.13),it yields

In(3.14),setting

We now apply Lemma 2.6 to(3.15).We can prove,respectively.

and

Substituting(3.18)and(3.19)into(3.17),

It is easy to see that there exists a positive constantθ<τsatisfying

It follows from(3.20)and(3.21)that

which implies